Nonlin. Processes Geophys., 19, 185–197, 2012 www.nonlin-processes-geophys.net/19/185/2012/ doi:10.5194/npg-19-185-2012 © Author(s) 2012. CC Attribution 3.0 License.
Nonlinear Processes in Geophysics
Particle trajectories beneath wave-current interaction in a two-dimensional field Y.-Y. Chen1, 3 , H.-C. Hsu2 , and H.-H. Hwung3 1 Dept.
of Marine Environment and Eng., National Sun Yat-Sen Univ., Kaohsiung 804, Taiwan Hydraulics Laboratory, National Cheng Kung Univ., Tainan 701, Taiwan 3 Dept. of Hydraulic and Ocean Eng., National Cheng Kung Univ., Tainan 701, Taiwan 2 Tainan
Correspondence to: H.-C. Hsu (
[email protected]) Received: 25 October 2011 – Revised: 29 January 2012 – Accepted: 6 February 2012 – Published: 16 March 2012
Abstract. Within the Lagrangian reference framework we present a third-order trajectory solution for water particles in a two-dimensional wave-current interaction flow. The explicit parametric solution highlights the trajectory of a water particle and the wave kinematics above the mean water level and within a vertical water column, which were calculated previously by an approximation method using an Eulerian approach. Mass transport associated with a particle displacement can now be obtained directly in Lagrangian form without using the transformation from Eulerian to Lagrangian coordinates. In particular, the Lagrangian wave frequency and the Lagrangian mean level of particle motion can also be obtained, which are different from those in an Eulerian description. A series of laboratory experiments are performed to measure the trajectories of particles. By comparing the present asymptotic solution with laboratory experiments data, it is found that theoretical results show excellent agreement with experimental data. Moreover, the influence of a following current is found to increase the relative horizontal distance traveled by a water particle, while the converse is true in the case of an opposing current.
1
Introduction
The problem of nonlinear water waves propagating through areas containing tidal, ocean or discharge current is an important issue in marine environments. The interaction between these flows plays vital roles in many aspect of coastal and ocean engineering, for example forces due to such flow fields on fixed or floating offshore wind turbines, sediment transport, contaminant and nutrient dispersion. The phenomenon of wave-current interaction has been studied extensively since the 1970s. Several theoretical solutions for waves on currents with uniform or sheared profiles have been
well documented in the review articles of Peregrine (1976), Jonsson (1990) and Thomas and Klopman (1997). Reports are also available on experimental studies for combined wave and current covering various aspects of this problem (Brevik, 1980; Constantin and Strauss, 2004; Kemp and Simons, 1982, 1988; Thomas, 1981, 1990). Most previous theories dealing with wave-current interactions have employed the Eulerian description, in which the free surface fluctuations can be expressed in a Taylor series expansion relative to a fixed water level (i.e., the still water level). This implicitly assumes that the surface profile of a wave is a differentiable single-valued function. Unlike the Eulerian free surface, which is given as an implicit function, a Lagrangian surface is described through a parametric representation of the position of a particle. The use of Lagrangian coordinates yields the only known nontrivial exact solutions to the governing equations for gravity water waves (i.e., Gerstner’s solution for deep-water waves (Gerstner, 1802) and a recently found edge wave solution along a sloping beach (Constantin, 2001) which was extended to stratified flows (R. Stuhlmeier, 2012)). The main advantage of such a description is to allow better flexibility for describing the actual shape of the ocean surface, which will be demonstrated later in this paper. Based on this reason, it has been shown that the Lagrangian description is more appropriate for the motion of the limiting free surface, which cannot be captured by the classical Eulerian solutions (Biesel, 1952; Chen et al., 2006; Naciri and Mei, 1992). However, reports on this notable improvement using Lagrangian description are rather limited. The first water wave theory in Lagrangian coordinates in which the flow possesses finite vorticity was presented by Gerstner (1802), which was re-discovered by Rankine (1863) and the modern discussion by Constantin (2001) and Henry (2008). Miche (1944) proposed perturbation Lagrangian solutions to the second order for a gravity wave. Pierson (1962) applied the Navier-Stokes equation to the
Published by Copernicus Publications on behalf of the European Geosciences Union & the American Geophysical Union.
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deep water waves in the Lagrangian formulae and obtained the first-order Lagrangian solution. Sanderson (1985) obtained second-order solutions for small amplitude internal waves in a Lagrangian coordinate system. Ng (2004) re-examined the problem of mass transport due to partial standing waves in one and two layer fluids. Buldakov et al. (2006) developed a Lagrangian asymptotic formulation up to the fifth order for nonlinear water waves in deep water. Clamond (2007) obtained a third-order Lagrangian solution for gravity waves in finite-depth water and a seventh-order solution for deep water waves. To date, only a limited few theoretical solutions are derived for wave-current interaction in Lagrangian coordinates. Umeyama (2010) gave a wrong third-order solution of particle trajectory, which did not include Lagrangian wave frequency and Lagrangian mean level shown in Eqs. (88) and (90) derived in this paper. Zaman and Baddour (2010) presented a first-order solution of particle trajectory in the combined wave-current flow. The theoretical investigations of the particle paths beneath a Stokes wave and solitary wave were recently undertaken by Constantin (2006, 2010). Figure 1 Definition This paper aims to study particle trajectories of a twodimensional wave-current field based on the fully Lagrangian current. framework, and to derive asymptotic solutions that can be used to describe the dynamics for the entire flow field. Previous works on progressive, standing, short-crested gravity waves and gravity-capillary waves have been summarized in the papers by Chen and Hsu (2009), Chen et al. (2010) and Hsu et al. (2010). In this paper, we look into the effect of uniform current on a gravity water wave, the motion of which is assumed to be inviscid, incompressible and irrotational. A set of governing equations in Lagrangian coordinates is derived for two-dimensional progressive gravity waves on uniform current in a constant water depth. We will construct asymptotic expansions of the solution in powers of the wave amplitude, which is assumed to be small using the Lindstedt-Poincare perturbation method. Approximate solutions including particle trajectory, Lagrangian wave period, the Lagrangian mean level and mass transport velocity are derived up to the third order. A detailed analysis of influences of the uniform current is then carried out. Finally, to validate the accuracy of the analytical results, a series of laboratory experiments are performed. The trajectories of water particles in a wave-current interaction flow are shown to have an excellent agreement with experimental data. The problem formulation and the procedures for constructing asymptotic solutions are described in Sect. 2. In Sect. 3, we derive equations for the properties of surface-particle trajectories and present results for some selected wave-current flow. Section 4 is devoted to a description of experimental apparatus and of the experimental procedure. In Sect. 5 the trajectories of surface and subsurface particles are presented. Some concluding remarks are given in the final section.
Nonlin. Processes Geophys., 19, 185–197, 2012
Fig. 1.showing Definition sketch showingofa system of progressive wave sketch a system progressive wave train on a train on a uniform current.
2
Formulation of the problem
We consider the problem of a two-dimensional monochromatic wave with a steady uniform current on an impermeable and horizontal bed (Fig. 1). The fluid motion is taken to be two-dimensional and irrotational, and the wave is rightgoing. We choose Cartesian axes with x pointing horizontally to the right and y vertically upward from the still water level. The mathematical problem is formulated in terms of Lagrangian variables, a and b, which define the original position of individual fluid particles. At any time t, we let b = 0 be the free surface, and b = d be the bottom. The Cartesian coordinates (x(a,b,t), y(a,b,t)) of fluid particles and the fluid pressure p(a,b,t) are the unknowns. Based on the Lagrangian description, the governing equations and boundary conditions for two-dimensional irrotational free-surface flow are summarized as follows: J=
∂(x,y) = 1, ∂(a,b)
(1)
∂J = xat yb + xa ybt − xbt ya − xb yat = 0, ∂t
(2)
xat xb − xbt xa + yat yb − ybt ya = 0,
(3)
∂φ = xt xa + yt ya , ∂a
(4)
∂φ = xt xb + yt yb , ∂b
p ∂φ 1 ∂x ∂y =− − gy + [( )2 + ( )2 ], ρ ∂t 2 ∂t ∂t
(5)
p = 0,
(6)
b = 0.
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Y.-Y. Chen et al.: Particle trajectories beneath wave-current interaction v = yt = 0,
(7)
y = b = −d.
In Eqs. (1)–(7), subscripts a, b, and t denote partial derivatives with respect to the specified variables, U denotes the speed of a steady uniform current, g is the gravitational acceleration, p(a,b,t) is water pressure, φ(a,b,t) is a velocity potential function in the Lagrangian system. Except for Eqs. (4) and (5), the fundamental physical relationships defining the equations above have been derived previously (Lamb, 1932; Naciri and Mei, 1992). Equation (1) is the continuity equation based on the invariant condition on the volume of a Lagrangian particle; Eq. (2) is the differentiation of Eq. (1) with respect to time. Equations (3) and (4) denote the irrotational flow condition and the corresponding Lagrangian velocity potential, respectively. Equation (5) is the Bernoulli equation for the irrotational flow in the Lagrangian description. The wave motion has to satisfy a number of boundary conditions at the bottom and on the free water surface. Equation (6) is the dynamic boundary condition of zero pressure at the free surface. On a rigid and impermeable bottom, the no-flux bottom boundary condition gives Eq. (7). 3
Asymptotic solutions
where the Lagrangian variables (a, b) are defined as the two characteristic parameters. In these expressions, fn , gn , φn and pn are expected to be associated with the nth-order harmonic solutions. fn0 , gn0 and φn0 are non-periodic functions that increase linearly with time. σ = 2π TL is the angular frequency of particle motion or the Lagrangian angular frequency for a particle reappearing at the same elevation. TL is the corresponding period of particle motion. Upon substituting Eqs. (8)–(12) into Eqs. (1)–(7) and collecting terms of equal order, we obtain a sequence of nonhomogeneous governing equations that can be solved successively, as shown in the following sections. 3.1
First-order approximation
Collecting terms of order ε, the governing equations and the boundary conditions can be obtained as follows: 0 + g + g 0 + [σ (f 0 f1a + f1a 1b 0a 1σ t + f1σ0 t ) 1b 0 +σ0b (g1σ t + g1σ0 t )]t = 0, 0 0 σ0 (f1aσ t + f1aσ + g1bσ t + g1bσ ) 0t 0t 0 0 +σ0a (f1σ t + f1σ0 t ) + σ0b (g1σ t + g1σ ) 0t 0 0 +σ0 {σ0a [f1(σ t)2 + f1(σ t)2 ] + σ0b [g1(σ t)2 + g1(σ
t)2
0 0 σ0 (f1bσ t + f1bσ − g1aσ t − g1aσ ) 0t 0t 0 ) − σ (g 0 +σ0b (f1σ t + f1σ + g 0a 1σ t 1σ0 t ) 0t 0 0 +σ0 {σ0b [f1(σ t)2 + f1(σ t)2 ] − σ0a [g1(σ t)2 + g1(σ
t)2
0
To solve the nonlinear Eqs. (1)–(7), we introduce the Lagrangian angular frequency σ of particle motion, which is a function of a nonlinear parameter and the Lagrangian level label (b), making the wave periodic in time t and space x (or a) in order to avoid a secular term. We use the LindstedtPoincare technique that yields uniform expansions to uncover the solutions in the Lagrangian system. The issue of convergence is covered by the recent regularity results in Constantin and Esher (2011). In the Lagrangian approach, the particle positions x and y, the potential function φ, and pressure p are considered as functions of independent variables a, b and time t. Following Chen and Hsu (2009), Chen et al. (2010) and Hsu et al. (2010), these solutions are sought in perturbation series by introducing an ordering parameter ε, which is inserted to identify the order of the associated term: ∞ X x = a +Ut + εn [fn (a,b,σ t) + fn0 (a,b,σ0 t)], (8)
0
y =b+
ε
[gn (a,b,σ t) + gn0 (a,b,σ0 t)],
(17)
p1 0 0 = U · σ0 (f1σ t + f1σ ) − σ0 (φ1σ t + φ1σ ) − g(g1 + g10 ), 0t 0t ρ
(18)
p1 = 0
(19)
at b = 0,
0 g1σ t = g1σ =0 0t
p = −ρgb +
∞ X
εn pn (a,b,σ t),
(10)
(11)
n=1
σ = σ0 (a,b) +
∞ X
ε σn (a,b) = 2π TL ,
n=1
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(12)
(20)
coshk(b + d) sin(ka − σ t), coshkd
sinhk(b + d) cos(ka − σ t), coshkd
(21a) (21b)
f10 = g10 = 0,
(21c)
σ0a = σ0b = 0,
(21d)
φ1 = ( n
on b = −d.
The flow is assumed periodic with a crest at a = 0 and t = 0, and hence the first-order solution can be easily written as
g1 = α
∞ X 1 φ = U a + U 2t + εn [φn (a,b,σ t) + φn0 (a,b,σ0 t)], 2 n=1
(15) ]}t = 0,
0 0 φ1b + φ1b + σ0b (φ1σ t + φ1σ )t 0t 0 0 0 = U · [(f1b + f1b ) + σ0b (f1σ t + f1σ )t] + σ0 (g1σ t + g1σ ), 0t 0t
(9)
n=1
0
(14) ]}t = 0,
(16)
f1 = −α n
0
(13)
0 0 φ1a + φ1a + σ0a (φ1σ t + φ1σ )t 0t 0 0 0 = U · [(f1a + f1a ) + σ0a (f1σ t + f1σ )t] + σ0 (f1σ t + f1σ ), 0t 0t
n=1 ∞ X
187
σ0 coshk(b + d) − U )α sin(ka − σ t), k coshkd
σ02 = gktanhkd,
(21e) (21f)
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p1 sinhkb = −gα cos(ka − σ t), ρ cosh2 kd
(21g)
where the parameter α represents the amplitude function of the particle displacement; the wave amplitude is, as usual, taken as a0 = αtanhkd, where k is the wave number (=2π/L, L is wave length). φ1 (a,b,t) is the first-order Lagrangian velocity potential and p1 (a,b,t) is the first-order wave dynamic pressure in the Lagrangian form with pressure p1 = 0 at the free surface b= 0. Equations (21a–g) satisfy all the hydrodynamic equations formulated in Lagrangian terms including the irrotational condition, and differ from Gerstner’s wave in infinite water depth, which possesses finite vorticity. The dispersion relation shows that the first-order Lagrangian wave frequency (σ0 ) is the same as that of the first-order Stokes wave frequency in the Eulerian approach (Biesel, 1952). The first-order free surface in Lagrangian coordinates is given by setting b = 0 in Eqs. (21a) and (21b), and is similar to expressions for the profile found from the first-order Eulerian equations. Equation (21d) is the basic velocity potential solution with a steady uniform current. In Eq. (21f), σ0 is the essential Lagrangian wave frequency for water particles relative to the uniform current. From this, it can be demonstrated that the Doppler’s effect is not apparent in the Lagrangian dispersion relation. This is correct: in Eulerian frame of reference, intrinsic wave frequency σ0 is different from absolute wave frequency (σ0 − kU ). 3.2
Second-order approximation
Collecting terms of order ε2 and using Eq. (21), the governing equations and the boundary conditions can be obtained as 0 0 f2a + f2a + g2b + g2b + f1a g1b − f1b g1a +(σ1a f1σ t + σ1b g1σ t )t = 0,
(22)
0 0 σ0 (f2aσ t + f2aσ + g2bσ t + g2bσ ) 0t 0t (23) +σ1 (f1a + g1b )σ t + σ0 (f1a g1b − f1b g1a )σ t +σ1a f1σ t + σ1b g1σ t + σ0 [σ1a f1(σ t)2 + σ1b g1(σ t)2 ]t = 0, 0 0 σ0 (f2bσ t + f2bσ − g2aσ t − g2aσ ) 0t 0t +σ1 (f1b − g1a )σ t + σ1b f1σ t − σ1a g1σ t +σ0 (f1a f1bσ t − f1aσ t f1b + g1a g1bσ t − g1b g1aσ t ) +σ0 [σ1b f1(σ t)2 − σ1a g1(σ t)2 ]t = 0, 0 = U · [(f + f 0 ) φ2a + φ2a 2a 2a 0 )t] + U · σ tf +σ0a (f2σ t + f2σ 1a 1σ t t 0 0 ) −σ0a (φ2σ t + φ2σ t 0 0 )+σ f +σ0 (f2σ t + f2σ 1 1σ t 0t +σ0 (f1a f1σ t + g1a g1σ t ) − σ1a tφ1σ t , 0 = U · [(f + f 0 ) + σ (f 0 φ2b + φ2b 2b 0b 2σ t + f2σ0 t )t] 2b 0 +U · σ1b tf1σ t − σ0b (φ2σ t + φ2σ0 t ) 0 +σ0 (g2σ t + g2σ ) + σ1 g1σ t 0t +σ0 (f1b f1σ t + g1b g1σ t ) − σ1b tφ1σ t ,
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(24)
(25)
p2 ρ
0 ) + g(g2 + g20 )] = −[σ0 (φ2σ t + φ2σ 0t 2 + g2 ) −σ1 φ1σ t + 12 σ02 (f1σ 1σ t t 0 ), 0 +U · σ0 (f2σ t + f2σ0 t ) + U · σ1 (f1σ t + f1σ 0t
and p2 = 0
at b=0.
0 g2σ t = g2σ = 0 on 0t
(28) b=−d.
(29)
Substituting Eqs. (21a∼g) into Eqs. (22)–(24), the secondorder governing equations in terms of ε2 , including the continuity equation and the irrotational condition, are given by 0 + g + g0 f2a + f2a 2b 2b t) 1 2 2 cosh2k(b+d) = 2k α ·{ + cos2(ka−σ } 2 2 cosh kd cosh kd coshk(b+d) −α · {σ1a · coshkd cos(ka − σ t) +σ1b · sinhk(b+d) coshkd sin(ka − σ t)} · t,
(30)
0 0 σ0 (f2aσ t + f2aσ + g2bσ t + g2bσ ) 0t 0t sin2(ka−σ t) 2 2 = α · k · σ0 · 2 cosh kd
−α · [σ1a coshk(b+d) coshkd cos(ka − σ t) +σ1b sinhk(b+d) coshkd sin(ka − σ t)]
(31)
−α · σ0 · [σ1a coshk(b+d) coshkd sin(ka − σ t) −σ1b sinhk(b+d) coshkd cos(ka − σ t)]t, 0 0 − g2aσ t − g2aσ ) σ0 (f2bσ t + f2bσ 0t 0t sinh2k(b+d) 2 2 = α k σ0 2 cosh kd
−α · [σ1b coshk(b+d) coshkd cos(ka − σ t) sinhk(b+d) −σ1a coshkd sin(ka − σ t)]
(32)
−α · σ0 · [σ1a sinhk(b+d) coshkd cos(ka − σ t) +σ1b coshk(b+d) coshkd sin(ka − σ t)]t. For gravity waves of permanent form, the terms t cos(ka − σ t) and t sin(ka −σ t) that increase linearly with time have to be zero to avoid resonance. Noting that σ1b = 0 or σ1 = ω1 = constant, then the general solution that satisfies the bottom boundary condition can be written as f2 = −β2
cosh2k(b + d)
cosh2 kd sin2(ka − σ t)
sin2(ka − σ t)
1 + α2k 4 cosh2 kd coshk(b + d) −λ2 sin(ka − σ t) coshkd 1 cosh2k(b + d) f20 = α 2 k σ0 t 2 cosh2 kd g2 = β2
(26)
(27)
1 + α2k 4
sinh2k(b + d) cosh2 kd sinh2k(b + d) 2
cosh kd
cos2(ka − σ t) + λ2
(33) (34)
(35)
sinhk(b + d) cos(ka − σ t) coshkd
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Y.-Y. Chen et al.: Particle trajectories beneath wave-current interaction g20 = 0.
(36)
Substituting Eqs. (33)–(36) into the irrotational Eq. (25) in ε2 order, we obtain φ2a = −2kUβ2
cosh2k(b + d) cosh2 kd
cos2(ka − σ t)
sinh2k(b + d) 2
cosh kd
λ2 = 0,
3 ω1 = 0,β2 = α 2 k(tanh−2 kd − 1), 8 (42)
The second-order Lagrangian solutions are assembled as
(37)
sin2(ka − σ t)
sinhk(b + d) sin(ka − σ t) coshkd sinh2k(b + d) +α 2 k 2 U σ0 t cosh2 kd sinh2k(b + d) +2σ0 β2 sin2(ka − σ t) cosh2 kd sinhk(b + d) sin(ka − σ t) +σ0 λ2 coshkd sinhk(b + d) +σ1 α sin(ka − σ t). coshkd −kU λ2
(38)
3 cosh2k(b + d) f2 = − α 2 k(tanh−2 kd − tanh2 kd) sin2(ka − σ t) 8 cosh2kd (43) 1 2 + α k(1 − tanh2 kd)sin2(ka − σ t), 4
1 cosh2k(b + d) f20 = α 2 k(1 + tanh2 kd) σ0 t, 2 cosh2kd
(44)
g2 = 38 α 2 k(tanh−2 kd − tanh2 kd) sinh2k(b+d) cosh2kd cos2(ka − σ t) + 14 α 2 k(1 + tanh2 kd) sinh2k(b+d) cosh2kd ,
(45)
g20 = σ1 = 0,
(46)
cosh2k(b+d) φ2 = σ0 −kU k β2 cosh2 kd sin2(ka − σ t) − 12 α 2 (σ0 − kU ) 12 sin2(ka − σ t) cosh kd − 14 α 2 kU 12 sin2(ka − σ t),
(47)
1 1 cosh2k(b + d) φ20 = α 2 σ02 (tanh2 kd − 1)t + α 2 kU σ0 t, 4 2 cosh2 kd
(48)
cosh kd
p2 ρ
Note that the secular terms in Eqs. (37) and Sect. 3.2 have to be eliminated. The second-order Lagrangian velocity potential is obtained by integrating over the Lagrangian variables a or b as σ0 − kU cosh2k(b + d) β2 sin2(ka − σ t) k cosh2 kd 1 1 1 − α 2 (σ0 − kU ) sin2(ka − σ t) 2 2 cosh2 kd 1 cosh2k(b + d) + α 2 kU σ0 t + D2 (σ0 t). 2 cosh2 kd φ2 =
(39)
and αω1 + λ2 (σ0 − kU ) = 0.
(40)
Substituting the solutions up to the second order into the energy Eq. (27) in ε2 order, we can get p2 ρ
cosh2k(b+d) 3 2 2 1 = {2σ0 σ0 −kU k β2 cosh2 kd − 4 σ0 α cosh2 kd −gβ2 sinh2k(b+d) + 2U σ0 β2 cosh2k(b+d) } · cos2(ka − σ t) cosh2 kd cosh2 kd σ0 −kU coshk(b+d) sinhk(b+d) (41) +{−gλ2 coshkd + σ1 k α coshkd coshk(b+d) coshk(b+d) +U σ0 λ2 coshkd + U σ1 α coshkd } · cos(ka − σ t) +{−σ0 D2σ0 t − 14 gα 2 k sinh2k(b+d) + 14 σ02 α 2 cosh2k(b+d) }. 2 2 cosh kd
Applying the zero pressure condition at the free surface, the unknown coefficients in Eq. (41) are obtained as
1 D2 = φ20 (σ0 t) = α 2 σ02 (tanh2 kd − 1)t. 4
1 1 cos2(ka − σ t) + α2k2U 2 cosh2 kd coshk(b + d) −λ2 kU cos(ka − σ t) coshkd cosh2k(b + d) +σ0 2β2 cos2(ka − σ t) cosh2 kd cos2(ka − σ t) −α 2 k cosh2 kd coshk(b + d) +(ασ1 + σ0 λ2 ) cos(ka − σ t), coshkd
0 φ2b + φ2b = −2kUβ2
189
cosh kd
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cosh2k(b+d) = {2σ0 σ0 −kU k β2 cosh2 kd sinh2k(b+d) 3 2 2 1 −gβ2 − 4 σ0 α cosh2 kd cosh2 kd +2U σ0 β2 } · cos2(ka − σ t) − 14 α 2 σ02 (tanh2 kd − 1) − 41 gα 2 k sinh2k(b+d) cosh2 kd + 41 α 2 σ02 cosh2k(b+d) . 2
(49)
cosh kd
The Lagrangian formulation for the particle trajectory at the second order approximation comprises a periodic component f2 and non-periodic function f20 . The latter increases linearly with time and is independent of the Lagrangian horizontal label a, which represents the mass transport, implying that a constant net motion would depend only on the vertical level b where the particle is located in the uniform current. The trajectory is the smallest near the bottom and is not a closed orbit as predicted by the first-order approximation. Moreover, Eq. (33), a second-order quantity, renders the same form obtained by Longuet-Higgins (1953) for the case of wave alone (i.e., without uniform current). The solution for vertical displacement g2 includes a second harmonic component and a time-independent term which is a function of wave steepness and the Lagrangian vertical label b. Overall, the expression of g2 yields vertical shift correction to a second-order which decreases with water depth. Taking the time average of the particle elevation g2 over a given period of a particle motion from Eq. (35), it can be shown that the mean level of water particle orbit in Lagrangian approach is higher than Nonlin. Processes Geophys., 19, 185–197, 2012
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that in the Eulerian counterpart, as suggested by LonguetHiggins (1979, 1986) for two-dimensional progressive water waves. For the limiting case U = 0, one can verify that the present theory reduces to pure progressive waves of constant depth, as was previously obtained by Chen et al. (2010). 3.3
Third-order approximation
The third-order governing equations and boundary conditions can be obtained by collecting the terms of 0 + g + g0 + f g + f g − f g f3a + f3a 1a 2b 2a 1b 1b 2a 3b 3b 0 )g + (σ f −(f2b + f2b 1a 2a 1σ t + σ2b g1σ t )t = 0,
(50)
0 0 ) + σ2 (f1aσ t + g1bσ t ) + g3bσ t + g3bσ σ0 (f3aσ t + f3aσ 0t 0t +σ2a f1σ t + σ2b g1σ t + σ0 [σ2a f1(σ t)2 +σ2b g1(σ t)2 ]t + σ0 [f1aσ t g2b + f1a g2bσ t (51) +f2aσ t g1b + f2a g1bσ t − f1bσ t g2a − f1b g2aσ t − (f2bσ t 0 0 )g +f2bσ )g1a − (f2b + f2b 1aσ t ] = 0, 0t 0 0 σ0 (f3bσ t + f3bσ − g3aσ t − g3aσ ) + σ2 (f1bσ t − g1aσ t ) 0t 0t +σ2b f1σ t − σ2a g1σ t + σ0 [σ2b f1(σ t)2 0 −σ2a g1(σ t)2 ]t + σ0 [f1a (f2bσ t + f2bσ ) 0t (52) +f2a f1bσ t − f2aσ t f1b 0 )+g g −f1aσ t (f2b + f2b 1a 2bσ t + g2a g1bσ t − g1b g2aσ t −g2b g1aσ t ] = 0, 0 = U · [(f + f 0 ) + σ (f 0 φ3a + φ3a 3a 0a 3σ t + f3σ0 t )t] 3a 0 +σ0 (f3σ t + f3σ0 t ) + σ2 f1σ t + σ0 [f1σ t f2a 0 )f + g +(f2σ t + f2σ 1a 1σ t g2a + g2σ t g1a ] − σ2a tφ1σ t , 0t 0 = U · [(f + f 0 ) φ3b + φ3b 3b 3b 0 )t] + (Uf +σ0b (f3σ t + f3σ 1σ t 0t 0 −φ1σ t )σ2b t + σ0 (g3σ t + g3σ ) 0t 0 +σ2 g1σ t + σ0 [f1σ t (f2b + f2b ) 0 )f + g +(f2σ t + f2σ 1b 1σ t g2b + g2σ t g1b ], 0t
(53)
(54)
= U · σ0 (f3σ t 0 )+U ·σ f +f3σ 1 2σ t + 0t U · σ2 f1σ t − [σ0 (φ3σ t 0 +φ3σ ) + g(g3 + g30 )] 0t 0 ) −σ2 φ1σ t + σ02 [f1σ t (f2σ t + f2σ 0t +g1σ t g2σ t ],
(55)
p3 = 0
(56)
0 g3σ t = g3σ = 0 on b=−d. 0t
From Eqs. (58)–(60), the secular terms that grow with time have to be zero. We can obtain sinh2k(b + d) σ2a = 0 and σ2b = −α 2 k 3 σ0 (61) . cosh2 kd Integrating Eq. (61) with b, σ2 is given by 1 cosh2k(b + d) σ2 = − α 2 k 2 σ0 + ω2 , 2 cosh2 kd
(62)
where ω2 is a constant which needs to be solved. Using Eq. (62), Eqs. (58)–(60) can be reduced to 0 + g + g 0 = αk 2 (2β f3a + f3a 3b 2 3b 1 2 cosh3k(b+d) cos(ka − σ t) + 4 α k) cosh3 kd +αk 2 (2β2 − 41 α 2 k) coshk(b+d) cos3(ka − σ t), 3
(63)
0 0 σ0 (f3aσ t + f3aσ + g3bσ t + g3bσ ) 0t 0t cosh3k(b+d) 1 2 2 sin(ka − σ t) = αk σ0 [(2β2 + 4 α k) cosh3 kd +(6β2 − 34 α 2 k) coshk(b+d) sin3(ka − σ t)], 3
(64)
0 0 σ0 (f3bσ t + f3bσ − g3aσ t − g3aσ ) 0t 0t 5 2 sinh3k(b+d) 2 = αk σ0 [(6β2 + 4 α k) cos(ka − σ t) cosh3 kd +(2β2 + 14 α 2 k) sinhk(b+d) cos3(ka − σ t)]. cosh3 kd
(65)
cosh kd
(57)
On substituting the first- and second-order approximations into the governing Eqs. (50)–(52), the third-order continuity, irrotational and energy equations become 0 + g + g0 f3a + f3a 3b 3b = αk 2 (2β2 + 14 α 2 k) cosh3k(b+d) cos(ka − σ t) cosh3 kd 1 2 coshk(b+d) 2 +αk (2β2 − 4 α k) cos3(ka − σ t) (58) cosh3 kd sinh2k(b+d) sinhk(b+d) 2 3 + σ2b ] · coshkd sin(ka − σ t) −α · {[α k σ0 cosh2 kd coshk(b+d) +σ2a coshkd cos(ka − σ t)} · t,
Nonlin. Processes Geophys., 19, 185–197, 2012
0 0 ) − g3aσ t − g3aσ σ0 (f3bσ t + f3bσ 0t 0t 3 2 sinh3k(b+d) 2 = αk σ0 [(6β2 + 4 α k) cos(ka − σ t) cosh3 kd 1 2 sinhk(b+d) cos3(ka − σ t)] +(2β2 + 4 α k) cosh3 kd + σ2b coshk(b+d) −α · {[ 21 α 2 k 3 σ0 sinhk(b+d) coshkd ]cos(ka − σ t) (60) cosh3 kd sinhk(b+d) −σ2a coshkd sin(ka − σ t)} −α · σ0 {[α 2 k 3 σ0 sinh2k(b+d) cosh2 kd +σ2b ] coshk(b+d) sin(ka − σ t) coshkd sinhk(b+d) +σ2a coshkd cos(ka − σ t)} · t.
cosh kd
p3 ρ
at b=0,
0 0 ) + g3bσ t + g3bσ σ0 (f3aσ t + f3aσ 0t 0t 1 2 cosh3k(b+d) 2 = αk σ0 [(2β2 + 4 α k) sin(ka − σ t) cosh3 kd +(6β2 − 34 α 2 k) coshk(b+d) sin3(ka − σ t)] cosh3 kd sinhk(b+d) sinh2k(b+d) 2 3 + σ2b ] coshkd sin(ka − σ t) (59) −α · {[α k σ0 cosh2 kd coshk(b+d) +σ2a coshkd cos(ka − σ t)} +α · σ0 {[α 2 k 3 σ0 sinh2k(b+d) + σ2b ] sinhk(b+d) coshkd cos(ka − σ t) cosh2 kd coshk(b+d) −σ2a coshkd sin(ka − σ t)} · t,
From Eqs. (63)–(65), the solutions of f3 , f30 , g3 and g30 can be assumed as f3 = [−β3 cosh3k(b+d) + 16 αk(5β2 cosh3 kd − 12 α 2 k) coshk(b+d) ]sin3(ka − σ t) cosh3 kd 1 2 −[ 2 αk(5β2 + α k) cosh3k(b+d) cosh3 kd +λ3 coshk(b+d) ]sin(ka − σ t), 3
(66)
cosh kd
g3 = [β3 sinh3k(b+d) − 12 αkβ2 sinhk(b+d) ]cos3(ka − σ t) cosh3 kd cosh3 kd 1 1 2 sinh3k(b+d) +[ 2 αk(3β2 + 2 α k) (67) cosh3 kd +λ3 sinhk(b+d) ]cos(ka − σ t), 3 cosh kd
www.nonlin-processes-geophys.net/19/185/2012/
Y.-Y. Chen et al.: Particle trajectories beneath wave-current interaction f30 = g30 = 0,
(68)
where β3 and λ3 are undetermined coefficients which can be found by using the dynamic free surface boundary condition. Substituting these terms, the first- and the second-order solutions into Eqs. (53) and (54), we can get 0 = U · [−3kβ cosh3k(b+d) φ3a + φ3a 3 cosh3 kd ]cos3(ka − σ t) + 21 αk 2 (5β2 − 21 α 2 k) coshk(b+d) cosh3 kd cosh3k(b+d) 1 2 2 −U · [ 2 αk (5β2 + α k) cosh3 kd coshk(b+d) ]cos(ka − σ t) +kλ3 cosh3 kd − 21 αk(3β2 − 12 α 2 k) coshk(b+d) ] +3σ0 · [β3 cosh3k(b+d) cosh3 kd cosh3 kd ·cos3(ka − σ t) +σ0 · [ 12 αkβ2 cosh3k(b+d) + (α · ωσ02 cosh3 kd +λ3 · sech2 kd) coshk(b+d) ] · cos(ka−σ t), 3
(69)
αω2 + σ0 λ3 sech2 kd = 0,
Finally, the physical parameters to the third-order solutions in Lagrangian form are given as follows:
cosh kd
(70)
cosh kd
f30 = g30 = φ30 = 0,
and (72)
The wave pressure can thus be given by σ2
= {β3 [3 k0 cosh3k(b+d) cosh3 kd 1 2 coshk(b+d) 2 −g sinh3k(b+d) ] − 2 α(7β2 − α k)σ0 cosh3 kd cosh3 kd + 21 αkgβ2 sinhk(b+d) } · cos3(ka − σ t) cosh3 kd 3 0 −σ0 D3σ0 t + { 2 αβ2 σ02 cosh3k(b+d) cosh3 kd − 21 αkg(3β2 + 12 α 2 k) sinh3k(b+d) cosh3 kd +(σ0 αk ω2 · cosh2 kd − 14 α 3 kσ02 ) coshk(b+d) cosh3 kd −gλ3 sinhk(b+d) } · cos(ka − σ t). 3
cosh kd
cosh2k(b + d) 1 σ2 = − α 2 k 2 σ0 2 cosh2 kd 1 2 2 + α k (9tanh−2 kd − 10 + 9tanh2 kd) · σ0 , 16
(80)
σ2
= {β3 [3 k0 cosh3k(b+d) − g sinh3k(b+d) ] cosh3 kd cosh3 kd coshk(b+d) 1 2 2 − 2 α(7β2 − α k)σ0 cosh3 kd + 12 αkgβ2 sinhk(b+d) } · cos3(ka − σ t) cosh3 kd cosh3k(b+d) 3 2 +{ 2 αβ2 σ0 cosh3 kd 1 − 2 αkg(3β2 + 21 α 2 k) sinh3k(b+d) cosh3 kd +(σ0 αk ω2 · cosh2 kd − 14 α 3 kσ02 ) coshk(b+d) cosh3 kd −gλ3 sinhk(b+d) } · cos(ka − σ t). 3
(81)
cosh kd
(73)
cosh kd
The procedure to obtain the solutions at this order is similar to that of O(ε2 ). The secular terms that grow with time have www.nonlin-processes-geophys.net/19/185/2012/
(78)
+ 61 αk(5β2 − 12 α 2 k) coshk(b+d) ] φ3 = U · [−β3 cosh3k(b+d) cosh3 kd cosh3 kd cosh3k(b+d) 1 2 ·sin3(ka − σ t) − U · [ 2 αk(5β2 + α k) cosh3 kd +kλ3 coshk(b+d) ] · sin(ka − σ t) cosh3 kd (79) + σk0 β3 cosh3k(b+d) sin3(ka − σ t) 3 cosh kd − 12 α(3β2 − 21 α 2 k) · σ0 · coshk(b+d) sin3(ka − σ t) cosh3 kd cosh3k(b+d) 1 + 2 αβ2 σ0 sin(ka − σ t), 3
p3 ρ
ω2 + λ3 · sech2 kd=0. σ0
(76)
g3 = [β3 sinh3k(b+d) − 12 αkβ2 sinhk(b+d) ]cos3(ka − σ t) cosh3 kd cosh3 kd 1 1 2 sinh3k(b+d) +[ 2 αk(3β2 + 2 α k) (77) cosh3 kd +λ3 sinhk(b+d) ]cos(ka − σ t), 3
φ3 = U · [−β3 cosh3k(b+d) + 16 αk(5β2 − 12 α 2 k) coshk(b+d) ] cosh3 kd cosh3 kd cosh3k(b+d) 1 2 ·sin3(ka − σ t) − U · [ 2 αk(5β2 + α k) + cosh3 kd kλ3 coshk(b+d) ] · sin(ka − σ t) cosh3 kd (71) + σk0 β3 cosh3k(b+d) sin3(ka − σ t) cosh3 kd coshk(b+d) 1 1 2 sin3(ka − σ t) − 2 α(3β2 − 2 α k)σ0 cosh3 kd + 12 αβ2 σ0 cosh3k(b+d) sin(ka − σ t). 3
α·
cosh3k(b + d)
1 coshk(b + d) 1 + αk(5β2 − α 2 k) ]sin3(ka − σ t) 6 2 cosh3 kd cosh3 kd 1 coshk(b + d) 2 cosh3k(b + d) −[ αk(5β2 + α k) + λ3 ]sin(ka − σ t), 2 cosh3 kd cosh3 kd
From Eqs. (69) and (70), we get
φ30 = D30 (σ0 t),
(74)
From Eq. (74), we get the solutions of β3 ,λ3 and ω2
f3 = [−β3
cosh kd
p3 ρ
to be zero. Using zero pressure condition at the free surface (p3 = 0 at b=0), we obtain
1 3 2 α k (9tanh−4 kd − 22tanh−2 kd + 13), β3 = 64 1 3 2 λ3 = − 16 α k (9tanh−2 kd − 10 + 9tanh2 kd)cosh2 kd, (75) 1 2 2 ω2 = 16 α k (9tanh−2 kd − 10 + 9tanh2 kd) · σ0 , D30 (σ0 t) = arbitrary constant = 0.
cosh kd
0 = U · [−3kβ sinh3k(b+d) φ3b + φ3b 3 cosh3 kd + 16 αk 2 (5β2 − 21 α 2 k) sinhk(b+d) ]sin3(ka − σ t) cosh3 kd sinh3k(b+d) 3 2 2 −U · [ 2 αk (5β2 + α k) cosh3 kd +kλ3 sinhk(b+d) ]sin(ka − σ t) cosh3 kd +3σ0 · [β3 sin3(ka − σ t) + 21 αkβ2 sin(ka − σ t)] · sin3k(b+d) cosh3 kd +σ0 · [− 12 αk(3β2 − 12 α 2 k)sin3(ka − σ t) +(α · ωσ02 cosh2 kd + λ3 )sin(ka − σ t)] sinhk(b+d) , 3
191
Equation (80) is the second-order angular frequency correction for a particle, in which the first term is the second-order Stokes wave frequency and the second term varies monotonically with the vertical label b or the wavelength-averaged level of the particles. The result differs from the Eulerian wave frequency. The third-order solutions of Eqs. (76) and (77) are periodic functions and have a combination of both Nonlin. Processes Geophys., 19, 185–197, 2012
192
Y.-Y. Chen et al.: Particle trajectories beneath wave-current interaction
first and third harmonic components. Thus, the solution of system has the following expressions: x = a + U t + εf1 (a,b,σ t) + ε2 [f2 (a,b,σ t) +f20 (a,b,σ0 t)] + ε3 f3 (a,b,σ t),
(82)
y = b + εg1 (a,b,σ t) + ε2 g2 (a,b,σ t) + ε3 g3 (a,b,σ t),
(83)
1 φ = U a + U 2 t + εφ1 (a,b,σ t) 2 +ε2 [φ2 (a,b,σ t) + φ20 (a,b,σ0 t)] + ε3 φ3 (a,b,σ t),
(84)
2
3
p = −ρgb + εp1 (a,b,σ t) + ε p2 (a,b,σ t) + ε p3 (a,b,σ t), 2
σ = σ0 (a,b) + ε σ2 (a,b) = 2π TL
(85) (86)
The set of Eqs. (82)–(86) ensures that Bernoulli’s condition of constant pressure is satisfied on the free surface. 4
parameters, namely the current velocity U, the wave height H, and wave period TE are reported along with other Lagrangian quantities parameters which were the mean value of three different measurements (the particle motion period TL , mass transport velocity and Lagrangian mean level). It shows an excellent agreement for the Lagrangian properties of water particle (Lagrangian wave frequency, mass transport and Lagrangian mean level) between the present third-order solution and experimental data. 5 5.1
Nonlin. Processes Geophys., 19, 185–197, 2012
Mass transport velocity
Taking time-average over one Lagrangian wave period to the terms of the horizontal particle displacement, the so-called drift velocity, over the whole range of depths can be obtained as follows:
Experimental setup and results
The aim of this experiment is to quantitatively investigate the characteristics of the water particle for periodic progressive gravity waves in uniform water depth. The experiments of particle trajectory beneath wave-current interaction have been carried out at the hydraulic laboratory of the Department of Marine Environment and Engineering of National Sun Yat-Sen University. The experimental setup, which comprises two interacting systems (1) a flume for the wave generation and propagation and (2) a recirculating apparatus, allows for a current to be generated. The produced current is nearly uniform with a variation of ±6.7 %. The wave flume is 35 m long, 1 m wide and 1.2 m high, with a fixed horizontal bottom. The waves are generated by means of a piston wavemaker, which is driven by a pneumatic system and is electronically controlled. The surface elevation is measured by means of several resistance wave gauges. An electromagnetic current meter (ACM-200A) is used to measure the current velocity. A camera was set up in front of the glass wall about 9.0 m from the wave generator to capture the particle motion. Four wave gauges were located at 7.0 m, 15 m, 16 m and 16.6 m from the wave generator to measure the incident waves. At the end of the tank, a 1:10 slope rubberizedfiber wave-absorbing beach was built to prevent the reflected wave. The orbital experiments (polysterne (PS) particle with diameter about 1mm and density near 1.05 g cm−3 ) were conducted at two constant water depths d (50 cm and 80 cm) and the various wave periods TE (0.96–2.06 s). The wave height H , which was the mean crest-to trough wave height computed over 20 different waves after the generated progressive waves became stable (about 7 waves), was varied over a range of about 3.17–15.2 cm. The particle motion was measured at different positions from the still water level to about 10.5 cm depth. In Table 1, the values of the control
Results and discussions
U+ x¯t c0
3 P
ε n [fnt (a,b,σ t)+fnt0 (a,b,σ0 t)]
n=1
= c0 = U + 12 α 2 k 2 (1 + tanh2 kd) cosh2k(b+d) cosh2kd ,
(87) c0 =
σ0 k ,
where the overbar denotes time-average over a Lagrangian wave period, i.e., the period of particle motion, where c0 is the linear phase speed. The second term of Eq. (87) on the right-hand side, a second-order correction quantity, is the same as that obtained by Longuet-Higgins (1953) as U = 0. From Eq. (87), the mass transport velocity is a function of the wave steepness, the water depth, uniform current and the vertical Lagrangian label. Differentiating Eq. (87) with respect to the vertical Lagrangian label b shows that the secondorder drift velocity is always positive but monotonically decays with depth from the surface to the bottom. In Fig. 2, the mass transport velocity is plotted against the water depth and the uniform current in the wave-current interaction field. From Fig. 2, we can clearly see that uniform current has a significant effect on the drift velocity. It can be seen that the effect of increasing current velocity is generally to augment the magnitude and extent of the time-averaged drift velocity, thus resulting in large horizontal distance traveled by a particle compared with the case without uniform current. The increasing of following current is to enhance the magnitude of the drift velocity over the whole range of depths, and a significant amount of fluid that has been transported forward, on the contrary, the drift velocity decreases by the adverse current. It is also remarkable that under the same wave, subsurface particles travel slower and diminish rapidly with the vertical position below the free surface.
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Y.-Y. Chen et al.: Particle trajectories beneath wave-current interaction
193
Table 1. Experimental conditions and comparison of measured and theoretical results of the particle period TL , mass transport velocity UM = x¯t − U and Lagrangian mean wave level η¯ L . No
TE (s)
d (cm)
H (cm)
a b c d e f g h
0.99 1.00 1.39 2.06 0.96 0.96 1.66 0.93
50 80 80 80 50 50 50 50
4.62 13.20 15.20 5.82 5.23 6.81 4.20 3.17
b (cm)
U (cm)
H L
d L
0 0 0 0 0 0 −9.5 0
2.95 9.63 8.93 11.72 −5.94 −6.60 −7.14 −21.01
0.031 0.080 0.052 0.012 0.037 0.048 0.013 0.024
0.333 0.485 0.274 0.158 0.349 0.349 0.155 0.376
Measured
Theory
Measured
Theory
Measured
Theory
1.000 1.063 1.430 2.064 0.976 0.979 1.663 0.934
4.41 19.12 14.63 12.12 −4.03 −3.36 −6.79 −20.25
4.38 19.15 14.59 12.15 −3.98 −3.31 −6.83 −20.21
0.12 0.77 0.65 0.06 0.17 0.26 0.03 0.08
0.11 0.78 0.63 0.07 0.15 0.25 0.04 0.06
-0.2
0.2
-0.4
0.1
b/d
(TL-TE)/TE
0.3
U/c0=0
η¯ L (cm)
1.002 1.061 1.433 2.062 0.979 0.982 1.661 0.936
0.0
-0.6
UM (b)(cms−1 )
TL (s)
0.0
U/c0=0.1 U/c0=0.2 U/c0=-0.1
-0.8
kU/0=0
-0.1
kU/0=+0.1 kU/0=-0.1
U/c0=-0.2 -1.0 -0.3
-0.2
-0.1
0.0
0.1
k
0
0.2
0.3
0.4
0.5
0.6
-0.2 0.00
0.02
0.04
0.06
0.08
0.10
0.12
0.14
H/L
xt
Fig. 3. The relative ratio for water particle motion at the free surface Figure 3. The relative ratio for water particle motion at the free surface between between Lagrangian and Eulerian periods for three current condiFig. 2. The dimensionless mass transport velocity profile cx¯0t xatt the Lagrangian and Eulerian periods for three current conditions. tions. relative water depth d/L = 0.5 and the relative wave height Hc/L = 0 at the relative water Figure0.12.under The dimensionless mass transport velocity profile various current conditions.
depth d / L 0.5 and the relative wave height H / L 0.1 under various current 5.2 Lagrangian wave frequency conditions
The Lagrangian angular frequency σ up to third order can also be obtained as σ = σ0 + σ2 (b) = σ0 − 12 α 2 k 2 (1 + tanh2 kd) cosh2k(b+d) cosh2kd σo −σ0 λα3 sech2 kd.
(88)
Hence, a general Lagrangian wave period σ differing from the Eulerian wave period for all particles at different vertical level b can be obtained directly in the odd-order Lagrangian solutions. The difference between the Lagrangian frequency σ and the Eulerian wave frequency σE is σ − σE = − 12 α 2 k 2 (1 + tanh2 kd) cosh2k(b+d) cosh2kd σo , λ3 2 σE = σ0 − σ0 α sech kd,
(89)
where σE is the angular frequency computed by Stokes expansion in the Eulerian system. It can be shown that Eq. (89) www.nonlin-processes-geophys.net/19/185/2012/
calculates the resultant wave period in Lagrangian form in a combined flow field for all the water particles at different elevations within the fluid domain. This equation also indicates the frequency of particle motion near the surface is smaller than that at the subsurface. For water particle motion at the free surface, the relative ratio between the Lagrangian form TL = 2π/σL and Eulerian form TE = 2π/σE for three different current conditions is shown in Fig. 3, in which TL /TE is found to increase with a following current (positive U ), and to decrease in an opposing current (negative U ) for a given wave steepness H /L. This implies that for a coplanar flow the water particles near the surface move forward further over one wave cycle than those against an opposing flow. Moreover, TL is larger than TE , even with the wave alone (e.g., the case of U = 0 in Fig. 3). Nonlin. Processes Geophys., 19, 185–197, 2012
194 Lagrangian mean level
(a)
0.3 0.4 0.2
k
Averaging the particle elevation up to the third order over a given period of particle motion, the present theory gives the Lagrangian mean level η¯ L (b), which is higher than the Eulerian mean level η¯ E = 0 as Z 1 TL 1 sinh2k(b + d) η¯ L − η¯ E = ydt= α 2 k . (90) TL 0 4 cosh2 kd
0.3 0.1 0.2 0.0
k
5.3
Y.-Y. Chen et al.: Particle trajectories beneath wave-current interaction 0.4
Longuet-Higgins (1979, 1986) also showed that the Lagrangian mean level is higher than the Eulerian mean level for progressive water waves. However, his expression is applicable only to particles at the free surface and is the same as that given by the first term of Eq. (90) at b = 0.
0.1 -0.1 0.0 -0.2
0
4
6
-0.2 2
4
6 (a)
Wave profiles and water particle orbits
x = a +Ut +
n=1
(fn + fn0 ),
(gn + gn0 ).
(92)
n=1
Figure 4 provides a comparison of the wave profiles between Lagrangian and Eulerian solutions, both to a thirdorder approximation. The results reveal that the height of the wave increases against an opposing current (negative Fr = U/c0 ) and decreases on a following current (positive Fr ). In Figure 4a, the Eulerian wave profiles have anomalous bumps in the trough for the wave conditions tested, which may not be a realistic physical phenomenon for waves of constant form. On the other hand, the Lagrangian wave profiles have sharper crests and broader troughs, as well as exclude any artificial bumps at or near the trough. Clearly the thirdorder Lagrangian solution is more exact than the Eulerian solution of the same order for describing the shape of the gravity wave. In general, the surface profile is an unknown function in the Eulerian approach, and the boundary conditions at the free surface can only be satisfied in an approximately manner. However, the free surface in the Lagrangian description is represented explicitly by a parametric function for the particles. The advantage of using Lagrangian description is that it allows flexibility for capturing the actual shape and the wave kinematics above mean water level. Thus theoretically, Lagrangian solution can provide better prediction for the wave profile at a large Froude number than the Stokes’ expansion to the same order. The wave profiles depicted in Nonlin. Processes Geophys., 19, 185–197, 2012
10
12
0.5 0.7 0.3
k
y =b+
3 X
8
(a)
0.7
0.5 0.1 0.3 -0.1
k
the horizontal and vertical particle trajectories are 3 X
12
(b)
(91)
n∈I
10
kx
The most important characteristic of fluid motion described by the Lagrangian solution is the trajectories of particles which are represented by Eqs. (82) and (83). The parameter α can be determined by the wave height H defined as half the vertical distance between the wave crest and wave trough, in wave number k and the water depth d given. Hence, we have H = [g1 + g3 ]b=0,ka−σ t=2nπ , 2
8
kx
0
5.4
2
-0.1
0.1 -0.3 -0.1 -0.5
0 -0.3
2
4
6
8
10
12
kx
-0.5
2 6 8 12 (b)profiles Fig. 4. 0 A comparison on4 the wave for the10 Eulerian and kx Lagrangian solutions both to a third-order under different current Figure 4.(a)AFcomparison the wave profiles for Eulerian and conditions = −0.2. Wave conditions d/Lthe = 0.2 r = 0.3, (b) Fron and H /L = 0.1 (Solid line: third-order Lagrangian solution; dash(b) solutions to a third-order under different current conditions (a) F dotted line: both third-order Eulerian solution).
Figure 4. .AWave comparison on the wave Fr 0.2 d/L 0.2profiles and Hfor / L the 0.1Eulerian (Solid and line: L conditions
solutions to that a third-order under different conditions (a) F Fig. 4 alsoboth show they are symmetric with current respect to the Lagrangian solution; dash-dotted line: third-order Eulerian solution) crest line, which were recently proven to hold true for irrotaFr 0.2 . Wave conditions d / L 0.2 and H / L 0.1 (Solid line: tional waves
In Fig. 5a–h, waterdash-dotted particle orbits for different Lagrangian solution; line:plotted third-order Euleriancursolution) rent magnitudes exhibit variations in orbital patterns, both in shapes and sizes, as a function of its original elevation. As can be expected in Fig. 5, the orbital displacement based on a third order solution is non-closed for a pure progressive wave. The elongation or shortening of the orbits in the case with following or opposing current is apparent in Fig. 5a–h. Their orbital dimensions in the cases with positive U values reflect the magnitude of a following current, and the converse is true for the condition with opposing current. It can be seen in each of the orbits plotted that a water www.nonlin-processes-geophys.net/19/185/2012/
Y.-Y. Chen et al.: Particle trajectories beneath wave-current interaction (a) TE =0.99sec ,H=4.62cm ,d=50cm ,b=0cm , U=2.95cm/sec
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Fig. 5. Comparisons between the orbits of water particles obtained by the presented theory and those from the experimental measurements of the PS motions at different water levels b in the various experimental wave cases, where solid line is the theoretical result and point is the Figure 5a~5h Comparisons between the orbits of water particles obtained by the experimental data which the time interval between two adjacent points is TE /20, TE is the wave period.
presented theory and those from the experimental measurements of the PS motions at different water levels b in the various experimental wave cases, where solid line is the theoretical result and point is the experimental data which the time interval between www.nonlin-processes-geophys.net/19/185/2012/ Nonlin. Processes Geophys., 19, 185–197, 2012 two adjacent points is TE / 20 , TE is the wave period.
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particle advances a distance forward, which is commonly referred to as mean horizontal drift or mass transport in the direction of wave propagation. The water particle at the free surface (b = 0) travels fastest, whilst that in the interior of the fluid propagates slower. To the third-order approximation, the particle trajectory has non-closed orbit, irrespective of their initial mean locations. This confirms the theoretical results obtained in Constantin (2006) and Constantin and Strauss (2010) for waves of large amplitude. In the case of wave on a following current, the effect of increasing current speed is generally to augment the magnitude and extent of the time-averaged drift velocity, thus resulting in large horizontal distance traveled by a particle compared with the case without current. Again, the converse is true when a wave train encounters an opposing current, which retards the advancement of water particles compared to that without a current or with a following current. As the strength of an opposing current becomes comparable with the wave speed, the water particle at greater depths beneath the still water level is mainly transport by the opposing current in larger current velocity and the direction of particle movement becomes contrary to wave progression. Figure 5a–h show excellent agreements between the measured trajectory and the theoretical trajectory predicted by the present third order Lagrangian wave theory. This is also in agreement with the theoretical findings in Constantin and Strauss (2010).
6
Conclusions
A particle-specific description of irrotational finite-amplitude progressive gravity waves on a uniform current in water of uniform depth satisfying all the governing equations and the boundary conditions is presented. The new Lagrangian solution is obtained to the third order. It can be used not only to determine the wave properties available in the Eulerian solution, but also to get the trajectory, the period, the mass transport and the Lagrangian mean level of a water particle, which are not available from the Eulerian solution. In the Lagrangian solution to a second-order, the Lagrangian mean level of a particle orbit over its motion period is found to be higher than that of the Eulerian, and it also has a timedependent term referred to as the mass transport velocity, which is applicable to the entire flow field. The frequency associated with water particle motions in Lagrangian form differs from that of the Eulerian, and the former is a function of wave steepness, uniform current speed and the Lagrangian vertical marked label b for each individual particle that can be obtained directly based on the third-order solutions. From the trajectories of water particles resulting from wave-current interaction, it is found that particle displacement near the surface decreases due to its mass transport velocity is resisted by an opposing current. Again in the case with an opposing current, the water particle further beneath the still water level is mainly transported by the opNonlin. Processes Geophys., 19, 185–197, 2012
posing current in the direction against the progressive wave, especially with a current in the large Froude number Fr . In the cases with a following current, the effect of increasing current speed is generally to increase the magnitude of the time-averaged mass transport velocity since the current is in the same direction as the wave propagation, thus resulting in augmentation to the horizontal distance traveled by a particle. Finally, a set of experiments analyzing the Lagrangian properties of nonlinear wave-current interaction flow is conducted in the wave tank. It shows excellent agreement between the experimental and theoretical results, including particle trajectories, Lagrangian wave frequency, mass transport velocity and Lagrangian mean water level predicted by the present third-order Lagrangian solution. Acknowledgements. The authors thank the anonymous referees for invaluable comments. This work was supported in part by the NSC grants 101-2911-I-110-504, 101-2622-E-006-010-CC2 and 99-2923-E-110-001-MY3 in Taiwan. Edited by: C. Kharif Reviewed by: H. B. Branger and another anonymous referee
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